Examples

Here we will show some examples of using ZXCalculus.jl.

Clifford simplification

This example can be found in the appendix of Clifford simplification. Firstly, we build up the circuit by using push_gate!.

using ZXCalculus

function generate_example_1()
    zxd = ZXDiagram(4)
    push_gate!(zxd, Val(:Z), 1, 3//2)
    push_gate!(zxd, Val(:H), 1)
    push_gate!(zxd, Val(:Z), 1, 1//2)
    push_gate!(zxd, Val(:H), 4)
    push_gate!(zxd, Val(:CZ), 4, 1)
    push_gate!(zxd, Val(:CNOT), 1, 4)
    push_gate!(zxd, Val(:H), 1)
    push_gate!(zxd, Val(:H), 4)
    push_gate!(zxd, Val(:Z), 1, 1//4)
    push_gate!(zxd, Val(:Z), 4, 3//2)
    push_gate!(zxd, Val(:X), 4, 1//1)
    push_gate!(zxd, Val(:H), 1)
    push_gate!(zxd, Val(:Z), 4, 1//2)
    push_gate!(zxd, Val(:X), 4, 1//1)
    push_gate!(zxd, Val(:Z), 2, 1//2)
    push_gate!(zxd, Val(:CNOT), 3, 2)
    push_gate!(zxd, Val(:H), 2)
    push_gate!(zxd, Val(:CNOT), 3, 2)
    push_gate!(zxd, Val(:Z), 2, 1//4)
    push_gate!(zxd, Val(:Z), 3, 1//2)
    push_gate!(zxd, Val(:H), 2)
    push_gate!(zxd, Val(:H), 3)
    push_gate!(zxd, Val(:Z), 3, 1//2)
    push_gate!(zxd, Val(:CNOT), 3, 2)

    return zxd
end
ex1 = generate_example_1()

We can draw this ZX-diagram by using

using YaoPlots
plot(ex1)

the circuit of example 1 To simplify zxd, one can simply use

simplified_ex1 = clifford_simplification(ex1)

or explicitly use

zxg = ZXGraph(ex1)
simplify!(Rule{:lc}(), zxg)
simplify!(Rule{:p1}(), zxg)
replace!(Rule{:pab}(), zxg)
simplified_ex1 = circuit_extraction(zxg)

And we draw the simplified circuit.

plot(simplified_ex1)

the simplified circuit of example 1

Phase teleportation

This example is an arithmetic circuit from phase teleportation. We first build up the circuit.

using ZXCalculus, YaoPlots
function generate_example2()
    cir = ZXDiagram(5)
    push_gate!(cir, Val(:X), 5, 1//1)
    push_gate!(cir, Val(:H), 5)
    push_gate!(cir, Val(:Z), 5)
    push_gate!(cir, Val(:CNOT), 5, 4)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 1)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:CNOT), 5, 4)
    push_gate!(cir, Val(:Z), 4, 1//4)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 1)
    push_gate!(cir, Val(:CNOT), 4, 1)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:Z), 1, 1//4)
    push_gate!(cir, Val(:Z), 4, 7//4)
    push_gate!(cir, Val(:CNOT), 4, 1)
    push_gate!(cir, Val(:CNOT), 5, 4)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 3)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:CNOT), 5, 4)
    push_gate!(cir, Val(:Z), 4, 1//4)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 3)
    push_gate!(cir, Val(:CNOT), 4, 3)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:Z), 3, 1//4)
    push_gate!(cir, Val(:Z), 4, 7//4)
    push_gate!(cir, Val(:H), 5)
    push_gate!(cir, Val(:Z), 5)
    push_gate!(cir, Val(:CNOT), 4, 3)
    push_gate!(cir, Val(:CNOT), 5, 4)
    push_gate!(cir, Val(:H), 5)
    push_gate!(cir, Val(:Z), 5)
    push_gate!(cir, Val(:CNOT), 5, 3)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 2)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:CNOT), 5, 3)
    push_gate!(cir, Val(:Z), 3, 1//4)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 2)
    push_gate!(cir, Val(:CNOT), 3, 2)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:H), 5)
    push_gate!(cir, Val(:Z), 2, 1//4)
    push_gate!(cir, Val(:Z), 3, 7//4)
    push_gate!(cir, Val(:Z), 5)
    push_gate!(cir, Val(:CNOT), 3, 2)
    push_gate!(cir, Val(:CNOT), 5, 3)
    push_gate!(cir, Val(:H), 5)
    push_gate!(cir, Val(:Z), 5)
    push_gate!(cir, Val(:CNOT), 5, 2)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 1)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:CNOT), 5, 2)
    push_gate!(cir, Val(:Z), 2, 1//4)
    push_gate!(cir, Val(:Z), 5, 7//4)
    push_gate!(cir, Val(:CNOT), 5, 1)
    push_gate!(cir, Val(:CNOT), 2, 1)
    push_gate!(cir, Val(:Z), 5, 1//4)
    push_gate!(cir, Val(:Z), 1, 1//4)
    push_gate!(cir, Val(:Z), 2, 7//4)
    push_gate!(cir, Val(:H), 5)
    push_gate!(cir, Val(:Z), 5)
    push_gate!(cir, Val(:CNOT), 2, 1)
    push_gate!(cir, Val(:CNOT), 5, 2)
    push_gate!(cir, Val(:CNOT), 5, 1)
    return cir
end
ex2 = generate_example2()
plot(ex2)

the circuit of example 2

We can use phase_teleportation for reducing the number of T gates of a circuit without changing its general structure.

reduced_ex2 = phase_teleportation(ex2)
plot(reduced_ex2)

the reduced circuit of example 2

By using tcount,

tcount(ex2)
tcount(reduced_ex2)

we can see that the number of T gates has decreased from 28 to 8.

Other usages

In the previous sections, we introduced how to use ZXCalculus.jl for ZX-diagrams which represent quantum circuits. Sometimes, one may wish to use it for general ZX-diagrams. It is possible.

One can create a ZXDiagram by building up its Multigraph and other information. For example,

using ZXCalculus, YaoPlots, LightGraphs
g = Multigraph(6)
add_edge!(g, 1, 2)
add_edge!(g, 2, 3)
add_edge!(g, 3, 4)
add_edge!(g, 3, 5)
add_edge!(g, 3, 6)
ps = [0, 1, 1//2, 0, 0, 0]
v_t = [SpiderType.In, SpiderType.X, SpiderType.Z, SpiderType.Out, SpiderType.Out, SpiderType.Out]
zxd = ZXDiagram(g, v_t, ps)

Because the information of vertices locations of a general ZX-diagram is not provided, its plot will have a random layout.

We can manipulate zxd by using ZX-calculus Rules.

matches = match(Rule{:pi}(), zxd)
rewrite!(Rule{:pi}(), zxd, matches)